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# probability ~ poisson distribution??? watch

1. hi guys

i am really confused with how to do this question:

"certain metal bars are such that the probability of a flaw in any sections of length ∂x is, for sufficiently small ∂x, approximately equal to λ∂x (λ>0), the probabilities associated with different sections being independent."

"prove that the probability that a rod of length d is without a flaw is e^(-λd)"

i think it is the poisson distribution, but i don't know where to start.

i would be grateful if u could give me some pointers on where to go with the question, but not the answer

thanks
2. If X has a Poisson distribution with parameter L,

P(X=x)= e^(-L)*(L^x/x!)

If there are no flaws, what value for x do you use?
3. (Original post by davemarkey)
hi guys

i am really confused with how to do this question:

"certain metal bars are such that the probability of a flaw in any sections of length ∂x is, for sufficiently small ∂x, approximately equal to λ∂x (λ>0), the probabilities associated with different sections being independent."

"prove that the probability that a rod of length d is without a flaw is e^(-λd)"

thanks
Split the rod into n subintervals of length d/n. The probability of no flaw in any interval is 1-λd/n for large enough n. [Ignoring second order terms in 1/n.]

So the probability of no flaw at all is

(1-λd/n)^n

In the limit as n tends to infinity this gives

e^(-λd)
4. yeah, thanks for that

but i am now stuck on the next part!

"if a rod has no flaw, its strength is S1, and if it contains one or more flaws, its strength is S2, where S1 > S2."

"find the mean and standard deviation of the strength of rods of length d"???

how do i work out the mean?

does it what the expected value?
5. (Original post by davemarkey)
yeah, thanks for that

but i am now stuck on the next part!

"if a rod has no flaw, its strength is S1, and if it contains one or more flaws, its strength is S2, where S1 > S2."

"find the mean and standard deviation of the strength of rods of length d"???

how do i work out the mean?

does it what the expected value?
I think the answer will have to be an infinite sum involving S1, S2, .... unless they've given you formulas for S1, S2, ...

So work out the probability of one flaw (much like I did for 0), then of 2 etc.

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